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Queueing Theory

Definition

The British Standards Institution’s definition is: “The mathematical theory
used in the analysis of systems in which service is provided under conditions
of varying supply and demand”, Term number 21064.

More comprehensively the subject is concerned with the mathematical analysis
and solution of problems in which items requiring service or items providing
service occasionally stand idle, and in which it is required to specify either
the arrival rate or the service rate or both, to achieve the most satisfactory
style of operation, e.g. least cost operation of the total system.

Purpose

Mathematical techniques, including probability theory, used in operations
or operational research to identify, illustrate and, it is hoped, influence
the characteristics of queues, whether of people, materials, work-in-progress,
etc. The object is to find the best way of planning the sequence of events
so that bottlenecks can be avoided.

Explanation.

Queueing theory deals with problems of congestion in which two opposing pressures
must be balanced. These two pressures are the rate of joining the queue and
the rate of discharge from it. Solutions can be reached analytically and
in some instances by alternative methods (simulation, qv). The theory of
queues requires a long exposition in which somewhat complex mathematics are
used. There can be no brief treatment of the subject that will handle real
cases. The analytical treatment depends on the probability distribution used
to represent arrival and service rates and the queue discipline.

Illustration:

One of the simplest examples that could be chosen to illustrate queueing
theory is the case of issuing stores. Ideally, the two requirements are that
the storemen will never be idle for lack of customers and that the customers
will never have to wait. These two requirements are mutually exclusive –
the more one approaches 100 percent utilisation of storemen, the greater
the waiting time of the customers. On the other hand, in order to guarantee
that customers will never wait, the number of storemen must be increased
to infinity. Both classes of idleness involve costs. The object of queueing
theory is usually to minimise the overall cost.